Is there really no way to force Time Machine to do a second, independent backup?

Every backup is a full backup. Your destination has hard links to each file so that if you copy any snapshot to another volume or folder - you have a copy of every file. If you delete a file from one time interval - the other intervals lose no data. As implemented, there is no reason to store duplicate bits on e destination if a hard link suffices to fully document each file being backed up.

Use Disk Utility to create a new partition and point Time Machine to that.

If you want to do a mirrored backup use something like rsync or Carbon Copy Cloner.

Time machine's mission is to make incremental backups every hour so you can browse and restore old versions of your documents. It wouldn't be my choice for making a full mirrored backup of a hard drive.

Rsync is built in to OSX so the price is right and it can synchronize two folders or whole drives.

Good luck!

In your situation, I'd try doing a backup with Super Duper.

www.shirt-pocket.com/SuperDuper/

It will be a full backup, can go on any drive with enough free space for the backup, and takes you away from any possible issues with Time Machine.